Create a game board consisting of n adjacent squares. Here’s a board for n = 10:

Still with me? Good.

The rules of POP are rather straightforward.

Players alternate turns, placing either an O or a P in any unoccupied square.

The winner is the first player to spell the word POP in three consecutive squares.

I first learned this game using O’s and S’s and trying to spell SOS, but for young kids, O’s and P’s are much better… the accidental occurrences of POO and POOP add a certain je ne sais quoi. (But not as much as foreign phrases add to a sentence about feces.)

Alex and Eli played this game tonight on the board shown above. After six turns, the game was decided. (As you can see, an accidental POO occurred in squares 6‑8. I mean an accidental occurrence of the word POO, not an actual occurrence of POO itself. If the latter had happened, the game would have ended immediately, and I wouldn’t be writing about it now.) It was Alex’s turn, and he realized that he lost: playing either an O or a P in squares 3‑4 would give Eli the win, and playing either an O or P in squares 9‑10 would just delay the inevitable.

“So, what’re you gonna do?” I asked.

Alex added an O to the third square, shrugged, and handed the pencil to Eli.

A coward dies a thousand deaths; the valiant die but once.

In that game, Alex went first and lost. So an immediate question:

Will the second player always win when n = 10?

This then leads to follow-up questions:

Are there other values of n such that the second player has a winning strategy?

Are there any values of n such that the first player has a winning strategy?

Are there values of n for which neither player has a winning strategy?

If you’d like to play a game of POP, then head over to The Game of POP spreadsheet on Google Drive, email the link to your friend, and start adding O’s and P’s. Feel free to change the size of the game board, too! Just please be a sweetie — when you finish, clear all your letters, reset the size of the game board to 10 squares, and be sure all the directions are retained at the top of the page.

About MJ4MF

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.